$Q$-Laguerre spectral density and quantum chaos in the Wishart-Sachdev-Ye-Kitaev model

TL;DR

This paper analyzes the spectral density and quantum chaos in the Wishart-Sachdev-Ye-Kitaev model, revealing connections to Q-Laguerre polynomials, chaotic dynamics, and universality classes in random matrix theory.

Contribution

It introduces an analytical approach to the spectral density of WSYK models, linking moments to Q-Laguerre polynomials and identifying new universality classes in level statistics.

Findings

Spectral density matches Q-Laguerre polynomial predictions for certain parameters.

Low-energy excitations exhibit stretched exponential growth for odd q.

Level statistics indicate quantum chaos and universality class membership.

Abstract

We study the Wishart-Sachdev-Ye-Kitaev (WSYK) model consisting of two $\hat{q}$-body Sachdev-Ye-Kitaev (SYK) models with general complex couplings, one the Hermitian conjugate of the other, living in off-diagonal blocks of a larger WSYK Hamiltonian. The spectrum is positive with a hard edge at zero energy. We employ diagrammatic and combinatorial techniques to compute analytically the low-order moments of the Hamiltonian. In the limit of large number $N$ of Majoranas, we have found striking similarities with the moments of the weight function of the Al-Salam-Chihara $Q$-Laguerre polynomials. For $\hat{q} = 3, 4$, the $Q$-Laguerre prediction, with $Q=Q(\hat{q},N)$ also computed analytically, agrees well with exact diagonalization results for $30 < N \leq 34$ while we observe some deviations for $\hat q = 2$. The most salient feature of the spectral density is that, for odd $\hat{q}$, low-energy excitations grow as a stretched exponential, with a functional form different from that of the supersymmetric SYK model. For $\hat q = 4$, a detailed analysis of level statistics reveals quantum chaotic dynamics even for time scales substantially shorter than the Heisenberg time. More specifically, the spacing ratios in the bulk of the spectrum and the microscopic spectral density and the number variance close to the hard edge are very well approximated by that of an ensemble of random matrices that, depending on $N$, belong to the chiral or superconducting universality classes. In particular, we report the first realization of level statistics belonging to the chGUE universality class, which completes the tenfold-way classification in the SYK model.